All Questions
1,923 questions with no upvoted or accepted answers
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42
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Relate existence of common invariant subspace of $m$ matrices to reducibility of certain polynomial
Let's recall a linear algebra fact: Let $A$ be an $n\times n$ matrix over a field $K$ and $\chi_A(t)$ be its characteristic polynomial. Then if $\chi_A(t)$ is reducible, $A$ would have a proper ...
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99
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Unimodular matrices fixing $(1, 1, \cdots, 1)$
What is known about the subgroup of $GL(n, \mathbb Z)$ fixing (under left multiplication) the vector ${(1, 1, \cdots, 1)}^T$ ('T' denotes transposition). I'm particularly interested in the case $n = 5$...
- 376
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0
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67
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The minimum number of polynomial equations the components of linearly dependent vectors must satisfy
Context:
Consider $m<n$ vectors $v_1,\dotsc,v_m\in\mathbb{C}^n$ with complex components. We can study if they are linearly dependent by constructing the following matrices. First the $n\times m$ ...
- 344
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1
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563
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Follow up: Show that these vectors are linearly independent almost surely
I posted this question some time ago here. I started a bounty for it and received an answer which helped me a lot. However, I still have some issues I want to discuss regarding it. Unfortunately I can'...
- 344
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149
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L_q matrix inequality
The following arose out of studying $\ell_q$ Lewis weights. Let $P$ be a real $n \times n$ orthogonal projection matrix (i.e., $P$ is symmetric and $P^2 = P$) and let $W$ be the diagonal matrix ...
- 101
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41
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Iterative algorithm for obtaining similarity
Let $x_1,x_2,\ldots,x_M$ be $M$ non-negative variables. Moreover, assume that $f_m(x_m)=\frac{x_m}{1+\sum_{n}\beta_{n}^{(m)}x_n}$ be $M$ fractional functions with non-negative constants $\beta_{n}^{(m)...
- 287
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0
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202
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Ratio of maximum to minimum value
Let $y = X \beta + \epsilon$, where $y \in R^{n}$, $X \in R^{n \times p}$, $\beta \in R^{p}$ and $\epsilon \in R^{n}$. Let $X = USV^\top$ be the SVD of the $X$. Let $u_i$ be the rows of $U$, then ...
- 61
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0
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114
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Error bounds on the expansion of square root of matrix
I'm working on a problem and was lead to trying to find an approximation for the square root of a matrix. I came across a way of doing this using holomorphic functional calculus. However, my first ...
- 427
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93
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Linear independence of Wishart matrices
Let $W\sim W_n(I,d)$ be a real Wishart matrix of an identity covariance matrix and $d$ degrees of freedom, i.e., $W=XX^T$ for $X$ being an $n\times d$ matrix whose entries are i.i.d sampled from a ...
- 123
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190
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Does the functional square root of the cosine admit a vector-based interpretation?
In linear algebra, the cosine of the angle between two vectors $a$ and $b$ is defined as $$\cos(a,b) = \frac{\langle a, b \rangle}{||a||\cdot||b||} .$$
The functional square root of the cosine has at ...
- 4,829
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108
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Solutions to matrix equations in the non-negative integers
For an integer matrix $S$, and an integer vector $y$, I'm looking for solutions to $xS = y$ where the entries in $x$ are in the non-negative integers.
I've been doing this with Sage's mixed integer ...
- 101
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127
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Maximizing a vector after a series of matrix multiplications
Problem Statement
Let's say we have a set of $n\times n$ matrices $X=\{M_1,\ldots,M_r\}$ and weights of these matrices $\{w_1,\ldots,w_r\}$ along with a set of "initial vectors" $\{v_1,\...
- 509
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70
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Birkhoff's theorem for hypergraphs
Birkhoff's theorem says that, in a bipartite graph $G$ in which both sides have size $n$, any fractional matching of size $n$ can be presented as a convex combination of integral matchings of size $n$ ...
- 3,559
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61
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Combining quadratic and linear matrix terms into a quadratic term
Given
$$ C = AFA^T + A\bar{F} $$
where $A = [A_1 A_2]$, $F = \begin{bmatrix}
F_1 & F_2 \\
F_2^T & F_3
\end{bmatrix}$, $\bar{F} = 2 \begin{bmatrix}
\bar{F}_1 \\
\bar{F}_2
\end{bmatrix}$ such ...
- 1
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45
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Critical Growth of Dimension for Dense Cover by Linear Subspaces
Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that
For any sequence of distinct finite-dimensional ...
- 5,405
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236
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Eigenvectors of a matrix
Let $M$ be a square matrix of order $n\times d$. Let $\xi_{1},\dots,\xi_{n\times d}$ be an orthonormal basis of $\mathbb{R}^{n\times d}$ formed of eigenvectors of $M$. We have
$$\xi_{i}=(\lambda_1, 0,...
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166
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Is the free algebra over an operad an algebra over that operad?
I'm asking here this question I asked on MSE that got no answers.
Let $V$ be a dg-module and $P$ an operad. The free $P$-algebra on $V$ is defined by $P(V)=\bigoplus_{r=0}^\infty (P(r)\otimes V^{\...
- 519
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54
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Is there a method to find a vector that optimizes a Rayleigh quotient over a subspace?
Let $M\in\mathbb{C}^{n\times n}$ be an arbitary Hermitian matrix and let $E$ be a subspace of $\mathbb{C}^n$.
Is there a method to find vectors $y,z\in E$ such that
$$\dfrac{y^*My}{y^*y}=\sup_{x\in E\\...
- 596
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0
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141
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Stabilizers in the action of $\mathrm{GL}(n, \mathbb Z)$ on $\mathbb Z^n$
How can we calculate effectively the subgroups of $G: = \mathrm{GL}(n, \mathbb Z)$ which fix pointwise a given submodule $S$ of $\mathbb Z^n$ in the action of $G$ on $\mathbb Z^n$ by left ...
- 376
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86
views
Eigenvalue inequality
Let
$g(\boldsymbol{\theta},\boldsymbol{\theta_0}) = trace [
\boldsymbol{\Omega{(\boldsymbol{\theta})}}^{-1} \boldsymbol{\Omega{(\boldsymbol{\theta_0})}}]-ln[det(\boldsymbol{\Omega{(\boldsymbol{\theta}...
- 31
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149
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Upper-triangular matrices as union of centralizers of cyclic elements
Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z}
)$ such that $n\leq p$. Consider the set $U$ of upper-triangular matrices of $G$
having entries of $1$ on the diagonal. The ...
- 463
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0
answers
30
views
Signs of difference matrices (sum of submatrices)
Given matrix $A \in \mathbb{R}^{m \times n}$, are there any results related to its difference array
$$A^* \triangleq \left[sign(a_{i,j} + a_{r, s} - a_{r, j} - a_{i, s})\right]_{i<r, j<s}?$$
Or ...
- 75
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answers
45
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On full rank submatrices of a construction
Take two matrices $T_1$ and $T_2$ in $\mathbb Z^{n\times n}$ with entries uniformly in $[-b,b]\cap\mathbb Z$ at some $b>0$. The matrices will be of rank $n$ each with probability at least $1-\frac1{...
- 1,836
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91
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Image of Frobenius element under irreducible representation is diagonalizable
Let $K/ \mathbb Q$ be a Galois extension, and $\rho$ be an irreducible representation of the Galois group $Gal(K/ \mathbb Q)$. Consider an integer prime $p$ which doesn't ramify in $K$, and let $\...
- 739
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159
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How to solve a non-local self-consistent equation
I have been struggling lately with solving numerically an equation of the form:
$$ g(x\pm x_{0}) = F[ g(x) ] $$
where $g(x)$ is a matrix satisfying the condition $g(x\to\pm\infty)=0$. My question is ...
- 123
0
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answers
47
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"Probability" for a partitioned matrix to be singular
Let $A,B\in\mathbb{R}^{n\times n}$ be two nonsingular matrices with $A\ne B$, and consider the following partitioned matrix
$$
M:=\begin{bmatrix}AA^\top + BB^\top & A^\top \Delta_1 A + B^\top \...
- 2,712
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answers
101
views
Find occurrences of certain matrix inside a matrix
This problem occurred from my need to find all graphs with a certain topology inside a bigger one. I don't need the subgraphs but the graphs that have the exact topology I am searching.
We know for ...
- 101
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132
views
How can GL(n) acts on the determinant polynomial?
I'm reading Landsberg's paper, which provide an introduction to geometric complexity theory. At chapter 2 of this paper, the author defined the following objects:
Let $W = \mathbb {C}^{n^2}$, $det_n \...
- 47
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0
answers
50
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Generalized eigenvectors product
Let's consider a real square matrix $A$ with eigenvalues $\lambda_n$ and eigenvectors $\mathbb x_n$, i.e. $A \mathbb x_n = \lambda_n \mathbb x_n$.
Suppose there are some generalized eigenvectors $\...
- 141
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50
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restriction of a formula with matrix inverse multiplied by a vector
I'm trying to reproduce a proof from this paper but I'm stuck in one point (Lemma 6). The general subject is bayesian model for multi-armed bandit problem solved with Thompson sampling.
I think I ...
- 1
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33
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Find a complex matrix on a unit sub-spheres
I am new to optimization theory. I have a following question. For a given $X = [x_1 x_2 \ldots x_N] \in \mathbb{C}^{N \times N}$, where $x_i \in \mathbb{C}^{N\times 1}$ for $i \in \{1,\ldots,N\}$, $U =...
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0
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132
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Upper bound on the condition number of the product of a random sparse matrix and a semi-orthogonal matrix
Let $G \in \mathbb{R}^{n \times m}$ (m > n, m = O(n)) whose all entries are i.i.d. distributed as $\mathcal{N}(0, 1) * \text{Ber}(p)$. Let $V \in \mathbb{R}^{m \times n}$ be a fixed semi-orthogonal ...
- 109
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0
answers
144
views
A Riemannian manifold with a non-degenerate metric and an inner product $u_{\beta} u^{\beta}=1$
The question is: given a Riemannian manifold with a non-degenerate metric g and an inner product $u_{\beta}u^{\beta}=1$, is $\nabla_{\mu} (u_{\alpha}u_{\beta})=0$ without demanding the trivial ...
- 9
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votes
0
answers
81
views
Constructing set with maximal independent subset
What is the minimal $m$ such that there exists a set $A = \{a_1,...a_n\}$ of vectors : $a_i \in \{0,1\}^m$ ($n$ is given) such that every subset of vectors of size $k$ is independent, but only with ...
- 51
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0
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59
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A system of inequalities involving a skew-symmetric integer matrix
Which skew-symmetric integer matrices $S$ satisfy the following inequalities
$SV_i \ne z_iE_i$ for all $i = 1,\cdots, n$
where
$V_i$ denotes the column with integer entries such that the $i$-th ...
- 376
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0
answers
41
views
Orthogonality condition of symmetric matrix pencil
Let $P(\lambda)=\lambda M−L\in \mathbb{R}^{n \times n}$ be a matrix pencil with symmetric nonsingular matrix $M$ and $L$ is a weighted Laplacian matrix of a connected graph. Clearly $(0,1_n)$ is an ...
- 21
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84
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A permutation statistic and determinantal identity
I'm trying to read this paper, Total positivity, Grassmannians and networks by Postnikov (https://arxiv.org/abs/math/0609764) and I'm stuck on Lemma 5.1, which is essentially an identity about maximal ...
- 1
0
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0
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136
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Finding a specific solution to $X^T\Sigma X = D$
I'm looking to solve for a specific $X$ in the following equation:
$$X^T\Sigma X = D,$$
where $\Sigma \succ 0$, $D$ is a diagonal matrix with strictly positive entries, and all matrices are square. It ...
- 41
0
votes
0
answers
99
views
Link between eigenvalues of a symmetric matrix and a functional space
Let $f_1,\dots,f_n \in L^2(\mathbb{R},\mathbb{R})$ be $n$ mutually orthogonal functions with $\int f^2_i =1$ such that $|\{x \in \mathbb{R} | f_i(x) = 0\}| = 0$ for any $i \in \{1, \dots,n\}$. Does ...
- 31
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0
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179
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Not unique eigenvalues in singular value decomposition
I have the following problem: I have a matrix $M\in \mathbb{R}^{3\times 3}$ and I consider two SVD's $U_1DV_1^T$ and $U_2DV_2^T$ of $M$ with $D = \mathrm{diag}(\lambda_1,\lambda_1,\lambda_2)$. ...
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94
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Neat expresion for an anti-symmetric matrix
Fix a column vector $\pmb{v}$ and consider the cross product $\pmb{v}^T\times\pmb{x}^T$ for any column vector $\pmb{x}\in\mathbb{R}^3$. One can write
$$\pmb{v}^T\times\pmb{x}^T=A(\pmb{v})\pmb{x}$$
for ...
- 43.2k
0
votes
0
answers
287
views
Complexity of pseudo-inverse of random matrix
Assume that $\mathbf{A}_{M\times N}$ is a sparse complex matrix. Then, what is the complexity of computation of its pseudo inverse, i.e.,
$$\mathbf{A}^{\mathrm{H}}(\mathbf{A}\mathbf{A}^{\mathrm{H}})^{-...
- 287
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votes
0
answers
84
views
Relation between two matrices associated with a positive definite function
Let $f:\mathbb{R}^N \to \mathbb{R}$ be a positive definite function. Let $$g(h) = \int_{\mathbb{R}^N}f(x)f(h-x)\mathrm{d}x$$ Due to Bochner's and Parseval's theorems, $g$ is also a positive definite ...
- 698
0
votes
0
answers
255
views
Span of a nonlinear function
Fix vectors $x,y\in\mathbb{R}^d$ and a smooth function $\phi:\mathbb{R}\rightarrow \mathbb{R}$. Define $\phi^d: \mathbb{R}^d \rightarrow \mathbb{R}^d$ as applying $\phi$ entrywise (i.e. $\phi^d(x_1, ...
- 159
0
votes
0
answers
353
views
Spectral norm of difference of quadratic matrices restricted to a subspace
Say that we have two matrices $X$ and $Y$ of dimensions $(T \times N)$ with $N < T$ and $rank(X)=rank(Y)=N$. Furthermore, define a $(T \times k)$ dimensional matrix $D$ with $k<N$ and $rank(D)=k$...
- 41
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0
answers
96
views
Eigenvalues of the matrix obtained by letting some of the rows vanish, hoping for some inequality
Let $A$ be an $n \times n$ matrix. Let $A_k$ be the matrix obtained by keeping the first $k$ rows of $A$ fixed and substituting $0$ for the rows $k+1$ to $n$. To be precise, we write $A= [R_1...R_k, ...
- 1,467
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votes
0
answers
400
views
Comparison of two similarity matrices
English is not my first language, so please excuse any mistakes.
I'm working with two similarity matrices on the same data set: Suppose I have $n$ items, and I calculated the similarity of each item ...
- 1
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0
answers
83
views
Nullity of infinite matrix in row echelon form
For an $m \times n$ matrix $A$ in row echelon form, $\mathrm{nullity} (A)$ is equal to the number of columns that do not contain a pivot. Is this also true for an infinite matrix in row echelon form ...
- 283
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votes
0
answers
67
views
Singular values and the chromatic number
What relation, if any, is there between the singular values of the adjacency matrix ( or possibly incidence matrix) of a simple graph and its chromatic number. Typically, do we have Hoffmann type, or ...
- 2,089
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votes
0
answers
195
views
Paths in graphs as a vector space or matroid
If I have a simple graph $G$, and what to count the number of simple paths between two distinct vertices, can the paths be seen as independent sets of a vector space, or even somehow, a matroid? I ...
- 640